Steve,
Have you actually tried to find in Plato's writings the source of the
`Platonic' doctrines that Aristotle refers to in Books M and N of
_Metaphysics_? There is no source there. There is of course the
traditional view of Plato's unwritten doctrines; but the difficulty with
that is that Plato was writing up to the time of his death, and it is
unreasonable to suppose that what he was saying in lectures was in flat
contradiction to what he was writing. The so-called intermediates of
mathematicals are a case in point. Where in Plato's writings does one
find such things? Again, the difficulties that Aristotle finds with the
notion of the `idea` Two, for example, being itself a paradigm of a
two-element set (a set of pure units), whether valid or not, do not apply
to Plato's doctrine as we can read it. (As a matter of fact, they are not
valid at all, any more than were Frege's objections to that idea in late
19th century.) Aristotle's reference to the `indefinite dyad', which is
generally refered to Plato's `lecture on the Good`, indeed has its source
in Plato's writings, namely in the _Philebus_ (which is on the Good). It
refers to a wonderful account of magnitude (in the passages on peras and
a peiron), including intensive magnitudes, which Aristotle completely
misunderstood and replaced in his own writings by his piddling `mixture
of opposites'. Due to Aristotle's influence among the ignorant churchmen
in the middle ages, it wasn't until the 14th century that the
mathematical treatment of intensive magnitudes reached the level implicit
already in the _Philebus_.
In past postings, you have praised the _Posterior Analytics_. To my mind,
that work is transformation of the central view of Plato on exact
science, to be found in the passage on the `second best method' in the
_Phaedo_ and in the discussion of noesis in the passage on the Divided
Line in Book VI of the _Republic_, adopting it, unsuccessively, to
Aristotle's empiricism. The essential difference between Plato's and
Aristotle's view is that, for the latter, all knowledge not only begins
in perception of sensibles, but is essentially tied to it. In the
perception of Socrates we see by abstraction man, in that we see animal,
etc. It is an epistemology of classification---suitable to Aristotle's
biology. But more to the point, all truth is ultimately empirical truth:
I can abstract from my shirt its color; but that only means that when I
speak of the color, I am really only speaking of the shirt, but in the
restricted vocabulary of color. For Aristotle, geometry is to be
understood in just this way: when we speak of geometrical objects, we are
really speaking of sensible substance, but only in the vocabulary of
extension.
Plato understood that this abstractionist conception of mathematics would
not work: you can't abstract what isn't there. For example, no empirical
truth will support, via abstraction, the existence of incommensurable
line segments. (My own hypothesis is that it was this particular fact
which most guided Plato's conception of exact science---though there were
other influences, such as Hereclitus and Parmenides.) The most important
feature of Plato's view of science (so obvious to us that we miss it) is
the autonomy of reason. We do not abstract from the empirical; but from
the phenomenon we may (causally) come to entertain a certain kind of
structure. We look for the first principles (axioms) for this kind of
structure (by a dialectical process); and from that point on, all
reasoning is purely logical.
For Plato, these axioms are not literally true of the phenomena, they are
idealizations. (He says that they are true of the forms; but one should
understand this as meaning nothing more than that the axioms aren't
arbitrary; there is an idea of a certain kind of structure behind them.)
This distinction between abstraction and idealization is the critical
distinction between Plato and Aristotle on exact science. Plato clearly
wins. The problem with Aristotle's Post. Analy., discussed over and over
again, is that there is a tension between two notions of truth: derivable
syllogistically from the first principles and empirically true.
Ultimately, the problem is: how can we be certain that the first
principles are true (emopirically). For Plato, the two questions are
separate: truth in exact science means derivable from the axioms. The
question of why the exact science (mathematics) should have empirical
significance is treated seperately---as it should be.
Just as Aristotle's epistemology is the epistemology of classification,
the logic which supports it, the syllogistic, is the logic of
classification. For all the hype to the contrary (especially Heath), it
had nothing to do with mathematical reasoning. For centuries after people
spoke of the syllogistic method versus the geometric method. (As late as
Kant, this was the distinction between demonstration (mathematical
reasoning) and discursive reasoning (syllogistic).
Contemporary analytic philosophers like Aristotle: they see him as an
analytic philosopher like themselves, only not so accute. (They overllook
the fact that he actually knew something, such as the functions of some
biological systems.) They like to think of Plato as a primitive
Aristotle, but who didn't understand the real nature of universals.
(MIchael Hardy (1/14/98) is absolutely right in his criticism of this.)
This is entirely wrong. The essential difference, from the point of view
of f.o.m., is that Plato believed in pure mathematics---the autonomy of
reason---and Aristotle's philosophy can not accomodate it.
Steve, I had decided not to get involved in what is only a historical
issue: but you have gone too far!
Bill Tait
Pensioner